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Introduction to Mathematical Tools
Part
II MATHEMATICAL TOOLS
This part is devoted to an extensive presentation of the mathematical notions that have been introduced in the framework of fuzzy set theory. Chapter 1 provides the basic definitions of various kinds of fuzzy sets, set-theoretic operations, and properties. Lastly, measures of fuzziness are described. Chapter 2 introduces a very general principle of fuzzy set theory: the so-called extension principle. It allows one to "fuzzify" any domain of mathematics based on set theory. This principle is then applied to algebraic operations and is used to define set-theoretic operations for higher order fuzzy sets. Chapter 3 develops the extensive theory of fuzzy relations. Chapter 4 is a survey of different kinds of fuzzy functions. The extremum over a fuzzy domain and integration and differentiation of fuzzy functions of a real variable are emphasized. Fuzzy topology is also outlined. Categories of fuzzy objects are sketched. Chapter 5 presents Sugeno's theory of fuzzy measures. In this chapter the link between such topics as probabilities, possibilities, and belief functions is pointed out.
II MATHEMATICAL TOOLS
This part is devoted to an extensive presentation of the mathematical notions that have been introduced in the framework of fuzzy set theory. Chapter 1 provides the basic definitions of various kinds of fuzzy sets, set-theoretic operations, and properties. Lastly, measures of fuzziness are described. Chapter 2 introduces a very general principle of fuzzy set theory: the so-called extension principle. It allows one to "fuzzify" any domain of mathematics based on set theory. This principle is then applied to algebraic operations and is used to define set-theoretic operations for higher order fuzzy sets. Chapter 3 develops the extensive theory of fuzzy relations. Chapter 4 is a survey of different kinds of fuzzy functions. The extremum over a fuzzy domain and integration and differentiation of fuzzy functions of a real variable are emphasized. Fuzzy topology is also outlined. Categories of fuzzy objects are sketched. Chapter 5 presents Sugeno's theory of fuzzy measures. In this chapter the link between such topics as probabilities, possibilities, and belief functions is pointed out.